Abstract
Clifford-Legendre and Clifford–Gegenbauer polynomials are eigenfunctions of certain differential operators acting on functions defined on m-dimensional euclidean space \({\mathbb R}^m\) and taking values in the associated Clifford algebra \({\mathbb R}_m\). New recurrence and Bonnet-type formulae for these polynomials are provided, and their Fourier transforms are computed. Explicit representations in terms of spherical monogenics and Jacobi polynomials are given, with consequences including the interlacing of zeros. In the case \(m=2\) we describe a degeneracy between the even- and odd-indexed polynomials.
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Acknowledgements
The authors would like to thank the anonymous referee whose suggestions have improved this paper. HBG and JAH are supported by the Centre for Computer-Assisted Research in Mathematics and its Applications at the University of Newcastle. JAH is supported by the Australian Research Council through Discovery Grant DP160101537. Thanks Roy. Thanks HG.
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This work was completed with the support the University of Newcastle’s CARMA Priority Research Centre.
This article is part of the Topical Collection on Proceedings ICCA 12, Hefei, 2020, edited by Guangbin Ren, Uwe Kähler, Rafal Abłamowicz, Fabrizio Colombo, Pierre Dechant, Jacques Helmstetter, G. Stacey Staples, Wei Wang.
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Ghaffari, H.B., Hogan, J.A. & Lakey, J.D. Properties of Clifford-Legendre Polynomials. Adv. Appl. Clifford Algebras 32, 12 (2022). https://doi.org/10.1007/s00006-021-01179-8
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DOI: https://doi.org/10.1007/s00006-021-01179-8